Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory Rainer Kühne To cite this version: Rainer Kühne. Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory. 2018. hal-01936246 HAL Id: hal-01936246 https://hal.archives-ouvertes.fr/hal-01936246 Preprint submitted on 27 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory Rainer W. Kühne Tuckermannstr. 35, 38118 Braunschweig, Germany e-mail: [email protected] July 2, 2018 Abstract Hehl et al. [6] (Einstein-Cartan theory), Mes- siah [7] (non-relativistic quantum mechanics), I examine the groups which underly classical Bjorken and Drell [8] (relativistic quantum me- mechanics, non-relativistic quantum mechanics, chanics), Bjorken and Drell [9] (quantum elec- special relativity, relativistic quantum mechan- trodynamics), Taylor [10] (quantum flavourdy- ics, quantum electrodynamics, quantum flavour- namics), Politzer [11] and Marciano and Pagels dynamics, quantum chromodynamics, and gen- [12] (quantum chromodynamics), and Kühne eral relativity. This examination includes the [13] (quantum electromagnetodynamics). rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U(1), SU(2), and SU(3) gauge trans- 2 Classical Mechanics formations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that The space of classical mechanics is described Dirac spinors can be described within the frame- by the three-dimensional Euclidian space. The work of gravitation theory. scalar product of the three-vectors ai and bi is given by a · b = δ a b (2) Keywords ij i j Einstein-Cartan theory, quantum gravity, where energy-momentum tensor, Lorentz transfor- δij = diag (1; 1; 1) (3) mations, gauge invariance, quantum field denotes the Kronecker symbol. The vector prod- theories uct is given by 1 Introduction (a × b)k = "ijkaibj (4) where "ijk denotes the totally anti-symmetric I will present my argumentation in a didactic Levi-Civita symbol. The square of the infinites- way. So I will start at a very elementary level. imal line element is given by If not denoted otherwise then Latin indices run 2 from 1 to 3, Greek indices run from 0 to 3. I will ds = δijdxidxj (5) use the Einstein summation convention, where it is summed over all indices which appear twice. where dxi denotes the infinitesimal coordinate Moreover I will use the natural units difference of the two space-points xi and yi ¯h = c = "0 = 1 (1) dxi = lim (yi − xi) (6) yi!xi where ¯h = h=2π denotes the reduced Planck con- A rotation R around the rotation angle ' in the stant, c the speed of light, and "0 the electric field constant. Inner indices of matrices will be two-dimensional subspace is described by the or- dropped. thogonal rotation group SO(2) Readers who would like to learn more about R = exp(−i'D) the theories which will be mentioned in this cos ' − sin ' paper are recommended to the comprehensive = 2 SO(2) (7) works of Goldstein [1] (classical mechanics), sin ' cos ' Jackson [2] (classical electrodynamics), Kilmis- where, because of the Euler equation ter [3] (special relativity), Weinberg [4] and Mis- ner, Thorne and Wheeler [5] (general relativity), e−i' = cos ' − i sin ' (8) 1 the generator of the rotation is SU(2) is covered by the group SO(3). Note also that the commutators eq. (11) and eq. (19) dif- 0 −i D = (9) fer by a factor of two. The anti-commutator is i 0 fσi; σjg = 2δij (20) A rotation R around the rotation angle 'i in the three-dimensional space is described by the The multiplication of two three-vectors ai and rotation group SO(3) bi is given by R = exp(−i'iDi) 2 SO(3) (10) (σ · a)(σ · b) = (σi · ai)(σj · bj) where the generators Di of the rotation satisfy = (δij + i"ijkσk)aibj the commutator relation = a · b + iσ · (a × b) (21) [Di;Dj] = i"ijkDk (11) According to non-relativistic quantum mechan- ics both spin and isospin are invariant under A representation of the generators of the SO(3) global transformations of the group SU(2). If group is Ψ denotes a two-component Pauli-spinor, 'i the 0 0 0 0 1 three-component rotation angle vector, σi the Pauli matrices, and x a space-time point, then D1 = @ 0 0 −i A (12) 0 i 0 Ψ0(x) = exp(−i'iσi=2)Ψ(x) (22) 0 0 0 i 1 D2 = @ 0 0 0 A (13) If Ψ denotes a two-component iso-spinor, 'i the −i 0 0 three-component phase vector, and τi the Pauli 0 0 −i 0 1 matrices, then D3 = @ i 0 0 A (14) 0 0 0 Ψ0(x) = exp(−i'iτi=2)Ψ(x) (23) Note that the number of generators of the or- thogonal groups is given by 4 Special Relativity dim SO(n) = n(n − 1)=2 (15) The special theory of relativity is invariant un- der the semi-simple Poincare group. The pa- and that the generators of SO(n) are Hermitean. rameters of its translational part are time and three-position. The parameters of its rotational part are the rotation angle three-vector and the 3 Non-Relativistic Quantum three-component Lorentz boost. Mechanics The scalar product of the two four-vectors aµ and bµ is given by Classical angular momentum is continuous. In µ ν quantum physics orbital angular momentum is a · b = gµν a b (24) quantized in units of ¯h and intrinsic spin is quan- tized in units of ¯h=2. Intrinsic spin is described where gµν denotes the metric tensor. It is by the unitary group SU(2). Its generators are gµν = g (25) the three Pauli matrices σi. An often used rep- µν µ µ µ resentation of the Pauli matrices is g ν = gν = δν = diag (1; 1; 1; 1) (26) 0 1 In Minkowski coordinates the metric tensor is σ1 = (16) 1 0 represented by 0 −i σ2 = (17) i 0 gµν = diag (1; −1; −1; −1) (27) 1 0 σ3 = (18) The square of the infinitesimal line element is 0 −1 given by 2 µ ν The commutator of the group SU(2) is given by ds = gµν dx dx (28) where dxµ denotes the infinitesimal coordinate [σ ; σ ] = 2i" σ (19) i j ijk k difference of the two space-time points xµ and µ The groups SU(2) and SO(3) are local isomor- y dxµ = lim (yµ − xµ) (29) phic for angles 0 ≤ ' < 2π where the group yµ!xµ 2 A rotation around the z-axis by the rotation an- 5 Relativistic Quantum Me- gle ' is given by chanics 0 1 0 0 0 1 µ B 0 cos ' − sin ' 0 C The group which underlies the kinematics of a ν = B C (30) @ 0 sin ' cos ' 0 A relativistic quantum mechanics and relativistic 0 0 0 1 quantum field theory is the Poincare group. Isospin is invariant under global transformations A Lorentz boost along the x-axis by the speed v of the group SU(2). The generators are the is given by three Pauli matrices τ . The three Pauli matri- 0 1 i γ −βγ 0 0 ces σi of spin are generalized by the four Dirac µ B −βγ γ 0 0 C matrices γµ. An often used representation of the a ν = B C (31) @ 0 0 1 0 A Dirac matrices is 0 0 0 1 1 0 γ = (43) where 0 0 −1 β = v=c (32) 0 σ γ = i (44) i −σ 0 γ = 1=p1 − v2=c2 (33) i µ It is In general, a Lorentz transformation a ν of a four-position xµ is given by 0 γ = γ0 (45) µ µ ν i x0 = a ν x (34) γ = −γi (46) The Lorentz transformation of a four-derivative The commutator is µ @µ = @=@x is given by ν [γµ; γν ] = −2iσµν (47) @0µ = aµ @ν (35) which is the definition of the generalized Dirac Finally, a Lorentz transformation around the pa- matrices σµν . The anti-commutator gives rameter !αβ is given by µ fγµ; γν g = 2gµν (48) µ 1 αβ a ν = exp !αβI 2 O(1; 3) (36) 2 ν By using the representation eqs. (43) and (44) where a representation of the six generators of of the Dirac matrices and the representation the Lorentz group SO(1; 3) is eqs. (16), (17) and (18) of the Pauli matrices, the anti-commutator gives the representation eq. 0 1 0 −1 0 0 (27) of the metric tensor in Minkowski coordi- 1 0 0 0 I10 = B C (37) nates. Moreover it is B 0 0 0 0 C @ A 5 0 0 0 0 γ = γ5 = −iγ0γ1γ2γ3 (49) 0 1 0 0 −1 0 The multiplication of two four-vectors aµ and bµ 0 0 0 0 I20 = B C (38) is given by B 1 0 0 0 C @ A µ ν 0 0 0 0 (γ · a)(γ · b) = (γµa )(γν b ) µ ν 0 0 0 0 −1 1 = (gµν − iσµν )a b (50) 30 B 0 0 0 0 C The Lorentz transformation of a four-component I = B C (39) @ 0 0 0 0 A spin 1/2 Dirac spinor field Ψ(x) is given by 1 0 0 0 i 1 µν 0 0 0 0 0 1 Ψ0(x0) = exp − σ ! Iαβ Ψ(x) 4 µν 2 αβ 0 0 0 −1 I13 = B C (40) (51) B 0 0 0 0 C @ A which is a generalization of eq.